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accumulation point – axiom
accumulation point
Given a set X, an accumulation point of X is a point p (which may or may not be in X) such that every neighborhood of p contains points of X different from p. See also: limit point, perfect set.
actual infinite
An infinite set considered as a completed whole, e.g., the natural numbers in modern set theory. Aristotle distinguished the actually infinite from the potential infinite, and considered that only the latter was permissible to thought.
Related MiniText: Infinity -- You Can't Get There From Here...
a.e.
See: almost everywhere.
algebraic lattice
A lattice L is said to be algebraic if every element of L is a join of compact elements of L.
algebraic number
A real or complex number which is the root of a polynomial with rational coefficients. Numbers that cannot be so expressed are called transcendental numbers. The algebraic numbers are countable.
algebra of functions
Given a topological space X, an algebra of functions on X is a subset A of the space of all continuous functions on X such that for all functions f and g in X we have:- fg is in A,
- f + g is in A, and
- for all constants c, cf is in A
whenever f and g are in A.
algebra of sets
Given a set X, an algebra on X is a collection of subsets of X which is closed under finite unions and complements. Such an algebra always includes the empty set and X itself. An algebra may also be defined as a ring of sets which includes X. An algebra of sets which is closed under countable unions is called a s-algebra.
almost disjoint
Set Theory: If a is an infinite cardinal and x and y are subsets of a, then x and y are said to be almost disjoint of their intersection has cardinality less than a. An almost disjoint family is a subset A of the power set of a, such that any two distinct members of A of size a are almost disjoint. A maximal almost disjoint family is an almost disjoint family not properly contained in any other almost disjoint family. It is a theorem that if A is an almost disjoint family and |A| = a, then A is not maximal.
Cf. D-system.
almost everywhere
Given a measure m on a measure space X, a condition (e.g., continuity of functions, etc.) is said to hold almost everwhere if it holds on X – N, i.e., on the set difference of X and N, where N is a null set. This is often abbreviated by "a.e."
alternating series test
ARTICLE
A test for the convergence of a series. See the article for a complete description.
antichain
If X is a set with a strict partial order  , then a subset A of X is called an antichain if no x and y in A are related by .
Cf. chain.
antinomy
A paradox arising from formal mathematical axioms. An antinomy may lead to a strict contradiction, as in the Russell Paradox, or merely offend our intuition, as in the Banach-Tarski Paradox.
antisymmetric relation
A relation “~” on a set X is antisymmetric if, for all x and y in X, whenever x ~ y and y ~ x then x = y.
Cf. symmetric relation, asymmetric relation.
antitone function
A function that is order preserving and decreasing.
arithmetic
The mathematical theory of addition, subtraction, multiplication, and division of integers. Formally, arithmetic is usually axiomatized by the so-called Peano axioms, and the theory is then often referred to as PA (for “Peano arithmetic”).
Aronszajn tree
Set Theory: For a a regular cardinal, an a-Aronszajn tree is a tree T with |T| = a, such that every chain in T is of cardinality less than a. Every Suslin tree is an Aronszajn tree. There are no Aronszajn trees of height w.
Ascoli-Arzelŕ Theorem
If {X} is a compact Hausdorff space and F is an equicontinuous, pointwise bounded subset of the space C(X) of continuous functions on X, then F is totally bounded in the uniform metric and the closure of F in C(X) is compact. Also, if X is a s-compact, locally compact Hausdorff space and if (fn} is an equicontinuous, pointwise bounded sequence in C(X), then there exists a subsequence and an f in C(X) such that the subsequence converges to f uniformly on compact sets.
asymmetric relation
A relation “~” on a set X is asymmetric if, for all x and y in X, x ~ y implies that it is not the case that y ~ x.
Cf. symmetric relation, antisymmetric relation.
automorphism
An isomorphism of a space to itself, that is, a bijective function on a space that preserves relationships, operations, and/or all essential properties (continuity, order, etc.).
Cf. homomorphism, group automorphism, ring automorphism, field automorphism.
automorphism group
The automorphisms of any object or structure can be composed to produce new automorphisms of that object. Since functional composition is associative, and since automorphisms are by their nature invertible, the automorphisms of an object or structure form a group (with the trivial automorphism serving as the identity element). This group is called the automorphism group of that object, usually denoted by Aut(X) for some object X. Automorphism groups are very useful, as they yield information about the symmetries of objects at hand.
axiom
In formal mathematics, a formula or schema of formulas stipulated as true in the theory under discussion, i.e., assumed to be true at the outset, and so not requiring proof. Axioms are the counterpart in mathematics of suppositions, assumptions, or premises in ordinary syllogistic logic.
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